pointsuch

Pointsuch is a neologism used in mathematical writing and computational contexts to denote the set of points that satisfy a given predicate. The term origins lie in the common phrase “points such that,” and it is sometimes employed as a compact label for a point-set defined by a condition P.

Formally, for a domain D (such as Euclidean space R^n) and a predicate P(x) that returns true or false, Pointsuch denotes the set { x in D | P(x) }. This aligns with standard set-builder notation and is particularly convenient when describing families of points defined by constraints, equations, or inequalities.

Examples help illustrate the concept. If P(x) is x^2 + y^2 ≤ 1 in R^2, Pointsuch is the closed unit disk. If P(x) requires x1 + x2 = 1 with x1, x2 ≥ 0, then Pointsuch is the line segment connecting (1,0) and (0,1). In optimization and geometry, Pointsuch can represent feasible regions, loci, or any subset of interest defined by a predicate.

Properties of Pointsuch depend on the predicate P. The set is a subset of D, and its topological character (open, closed, bounded) is determined by the form of P. If P is continuous and defined by inequalities, the resulting Pointsuch may be closed; equalities can yield lower-dimensional manifolds.

Usage notes: the term is informal and not widely standardized. More common alternatives include “the set of points satisfying P” or “points that satisfy P.” In code or pseudocode, Pointsuch may appear as a variable representing such a set. See also set-builder notation, predicate, and locus.